tax credit. Although we have not discussed this case, a natural benchmark for the ex-day price drop for these observations is 1:1. Considerations familiar from the literature on the ex-day in the United States could lead to a lower price drop.37 Second, the tax credit rate dropped from 56.25% in 1993 to
42.86% in 1994. Third, the specific predictions about α and β obtain for a sufficiently large dividend yield; for small dividend yields we expect a reduction in the slope coefficient.
The first two considerations can be addressed in a single regression. Define the dummy variables Dcredit equal to 1 when a dividend has a credit, D94−
equal to 1 when the dividend precedes 1994, and D94+ equal to 1 when the
dividend is in 1994 or later. Consider a modified version of Equation (24): Pi, t− Pi, t+1 Pi, t + r CDAX t, t+1 = α0+ α1Dcredit + β0+β1DcreditD94−+ β2DcreditD94+Di, t Pi, t + ui, t. (25)
36Spot checks of reported volumes for the non-DAX 30 revealed that for these stocks, volume numbers are often either unavailable or small.
37As discussed in Section 1.5, there is no obvious clientele which should lead to a price drop of greater than one- for-one. The tension between individual investors and brokers, however, could lead to a less than one-for-one price drop.
Table 5
OLS estimates for DAX 30 and Other (non-DAX 30) stocks
No. of α0 β0 α0+ α1 β0+ β1 β0+ β2 β1− β2 observations R2 DAX 30 0.0086 −0.0761 −0.0051 1.2574 1.2319 0.0254 245 0.405 (0.0242) (1.2771) (0.0026) (0.1007) (0.1274) (0.0886) Other 0.0167 0.9443 −0.0065 1.2408 1.3231 −0.0823 1950 0.202 (0.0261) (0.8456) (0.0018) (0.0770) (0.0620) (0.0688) Estimates are of Equation (25):Pi, t −Pi, t+1Pi, t + rCDAX
t, t+1 = α0+ α1Dcredit+ β0+ β1DcreditD94−+ β2DcreditD94+
Di, t
Pi, t+ ui, t
using a 1-day (DAX 30) and 5-day (other) window to compute the ex-day drop. α0+ α1is the intercept for a dividend which
has a credit attached, in any year. β0+ β1is the slope coefficient for a pre-1994 dividend with credit, while β0+ β2is the slope
coefficient for a post-1994 dividend with credit. α0and β0are the intercept and slope for dividends with no credit. Standard
errors are in parentheses.
The regression assumes that dividends bearing a credit have the same inter- cept in all years, but allows for a change in the slope coefficient when the tax rate changes. α0 and β0 are the intercept and slope coefficient for creditless
dividends in any year. For a pre-1994 dividend with credit, the intercept is α0+ α1 and the slope coefficient is β0+ β1. For a post-1994 dividend with
credit, the slope coefficient is β0+ β2.
Table 5 reports the results from estimating Equation (25). The table reports those coefficients and combinations of coefficients that are relevant, along with standard errors.
Because of the small number of observations where there is no credit, the price-drop coefficient for those cases is imprecisely estimated. For both the DAX and non-DAX samples, the hypothesis that the intercept is 0 and the slope coefficient is either 0 or 1 cannot be rejected. More interesting are the coefficients reflecting the slope coefficient pre- and post-1994. For all cases the slope coefficients are in the range 1.23–1.32, and the hypothesis that the two coefficients are equal in a given regression cannot be rejected. The stock data thus does not identify the change in the tax credit rate between 1993 and 1994.
To address the issue of different ex-day behavior for stocks with small dividend yields, regressions similar to Equation (25) were run with a dummy variable for dividends below 1%. The results are not reported. The regression coefficients reflecting the effects of larger dividends were similar to those in Table 4. The coefficients for smaller dividends were imprecisely estimated, with the intercept not significantly different from 0 and the slope coefficient not significantly different from 0 or 1.
5.2 Futures
As discussed in Section 2.4.1, holders of the DAX index futures contract do not receive the tax credit, and it is not reflected in the calculation of the DAX index. From the no-arbitrage bounds in Equation (22), we can com- pute a minimum theoretical futures price. Suppose that transaction costs are
zero,38 and that Germans performing a cash-and-carry arbitrage (long index,
short futures) can receive the credit, hence ε = 1. From Equation (22), the minimum price is
F0, t = P0(1 + r0, t)−F VP[Dk]. (26)
The intuition for this expression is that the stock price reflects the receipt of the tax credit, but not the forward price. Thus the forward price is lower by the future value of tax credits not received over the life of the contract.
Using Equation (26), we can specify a regression that permits us to test whether the interest rate, dividends, and the credit enter the forward price as predicted. Let rt, T be the DM-denominated LIBOR interest rate over the period until expiration (computed using the interest maturity closest to match- ing the maturity of the futures contract), and F Vt, T(D) the future value of dividends, calculated using the same LIBOR rate, payable on DAX 30 stocks between time t and time T , the futures expiration.39 It is assumed that divi-
dends actually paid were expected to be paid. We then estimate the regression Ft, T Pt =α0+ α1D94 −+β0rt, T T − t T + β1+ β2D94− FVt, T(Dt) Pt +ut. (27) As before, D94−is a dummy variable that is 1 before 1994. As with stocks, we
permit the slope coefficient on dividends to vary with the change in the tax credit. From Equation (22), we expect that changes in the interest rate are on average reflected one-for-one in the futures price and that the intercept is one. Thus we expect α0= 1, α0+ α1= 1, and β0= 1. The dividend coefficients
will depend on whether there is tax risk associated with arbitrage, but in any event we expect 0 > β1> −.4286, and 0 > β1+ β2> −.5625.
Table 6 reports the results of estimating Equation (27). Results for the full sample (beginning on November 23, 1990) indicate that the futures price is reduced by approximately 55% of the tax credit. The coefficient on the interest rate, however, is significantly less than 1, which indicates that the theoretical futures pricing formula does not hold.40This is due to some char-
acteristic of the data during the first year of the contract.
38The transaction costs associated with index arbitrage are potentially quite low. According to one market participant, many arbitrages in the DAX are effected using index basket trades and an exchange of physical for futures (EFP), in which an investor can close out the futures contract by delivering the cash index directly to a counterparty. This avoids many of the usual costs of exchange and market transactions.
39It should be noted that because of seasonality in dividend payments there is significant variation in the dividend variable. For approximately 25% of the observations, the dividend yield is 0. The estimated pricing relationship is the same when these observations are excluded.
40The regression using the full sample also had positively serially correlated errors, assessed using the regression approach suggested in Davidson and MacKinnon (1993, pp. 357–358). The three regressions with early observations omitted, however, displayed no significant first-order serial correlation.
Table 6 OLS regression
No. of
Start date α0 α0+ α1 β0 β1 β1+ β2 observations R2
11/23/90 1.0007 0.9999 0.8972 −0.2491 −0.2959 1771 0.399 (0.0002) (0.0003) (0.0293) (0.0400) (0.0431) 11/26/91 1.0003 1.0000 0.9829 −0.2740 −0.3799 1522 0.402 (0.0002) (0.0004) (0.0344) (0.0408) (0.0525) 6/1/92 1.0002 1.0000 1.0058 −0.2806 −0.4878 1397 0.375 (0.0003) (0.0005) (0.0384) (0.0422) (0.0671) 11/26/92 1.0002 1.0001 1.0113 −0.2822 −0.5072 1272 0.299 (0.0003) (0.0005) (0.0459) (0.0434) (0.0720)
Estimates of Equation (27):Ft, TPt = α0+ α1D94−+ β0rt, T T −tT +(β1+ β2D94−)F Vt, T (Dt )Pt + ut. Dependent variable is the
futures premium, Ft, T/Pt, the daily closing futures price for the nearest-to-expiration DAX 30 futures contract, divided by the
cash value of the DAX 30 index. Independent variables are interest over the life of the contract (β0) and the future value of
the dividend yield over the life of the futures contract post-1993 (β1) and the future value of the dividend yield over the life of
the futures contract pre-1993 (β1+ β2). Expiration months are March, June, September, and December. Dividend yield on cash
index constructed by author. Data is daily from “start date” in the table to December 12, 1997, the expiration of the December 1997 futures contract. Standard errors are in parentheses.
Table 6 reports results with the first 250 observations (1 year), 375 obser- vations (1.5 years), and 500 observations (2 years) removed.41 In all three
cases the interest rate coefficient and all intercepts are insignificantly differ- ent from 1, thus the low interest rate coefficient for the full sample is due to the first year of data. Moreover, in two of the additional regressions, the estimated effect of the tax credit is significantly greater for the early than the late years, and insignificantly different from the tax credit of .5625 for the years prior to 1994. Point estimates suggest that between 65% and 89% of the credit is reflected in the futures price prior to 1994, and about 65% thereafter.
The results are consistent with the futures price obeying the theoretical pricing formula. Unlike with stocks, the 1994 change in the tax credit is apparent and prior to 1994, we cannot reject the hypothesis that the full tax credit is impounded into the futures price. This suggests that the tax risk of undertaking arbitrage transactions may have increased since 1993. In addition, if more of the tax credit was impounded in futures, it may be that prior to 1994, futures arbitrage was preferred to stock trading as a way to exploit the tax credit.
5.3 Options
Option prices should reflect the tax credit in the same way as the futures prices. In theory one should be able to examine a set of option prices and infer the volatility and dividend implicit in prices, yielding an implied tax credit. However, data on German equity options is only sporadically available from Datastream and appeared to be of variable quality. Attempts to obtain 41Regression results did not change appreciably when the data was started still later.
Table 7
American put and call prices for Daimler-Benz options
Put premiums Call premiums
May 27 May 25 May 26
Stock price = 197.7 Stock price = 195.6 Stock price = 201
Strike Premium Volume Strike Premium Volume Premium Volume
200 32.26 1248 140 57.63 3503 58.51 7715 210 41.85 0 150 47.63 3658 48.52 13125 220 51.71 0 160 37.63 1603 38.52 6219 230 61.59 2178 170 27.63 2251 28.52 2371 240 71.51 238 180 17.63 1607 18.52 12642 250 81.50 23686 190 8.17 8216 8.52 3547
Prices are for options on May 25, May 26, and May 27, 1998. Daimler-Benz was to pay a DM 21.6 dividend with an ex-date of May 28. Options expired June 19. Source: Datastream.
reliable implied volatility and implied dividend estimates using this data had only mixed success.42
Fortunately, however, Daimler-Benz in May 1998 paid an unusually large dividend of DM 21.60 on a share price of DM 197.7, providing an unam- biguous example of the tax credit being embedded in option prices. This one example does not provide a precise point estimate of γ , but it does make clear the tax credit effect in option prices.
Table 7 displays put and call prices for June 1998 Daimler-Benz options, just prior to the May 28 ex-date. Exercise of American-style stock options in Germany is not permitted on the last cum-dividend day, hence May 26 is the last day on which a call can be exercised cum-dividend.43Puts are unaffected
since they are optimally exercised ex-dividend.
Reported call prices are close to intrinsic value on May 25, and slightly below intrinsic value on May 26, suggesting that early exercise prior to the dividend is expected.44 The in-the-money call prices provide no information
about the size of the dividend.
Puts are optimally exercised ex-dividend, thus their premiums reflect the ex-dividend value of the stock. Without considering the tax credit, the theo- retical upper bound for the price of an American put with strike price E and time to expiration T is [see Cox and Rubinstein (1985)]
P ut(E, T ) ≤ Call(E, T ) − P + E + D,
where P is the stock price and D is the present value of dividends remaining over the life of the option.
42Potential problems include data errors (reporting errors, bid/ask effects, stale prices) and the use of a pricing model which does not account for volatility skew. There were also occasional obvious data problems such as reported option prices below intrinsic value.
43I am grateful to a trader at Hull Trading for clarifying this point.
44Of course, American option prices below intrinsic value should not be observed. The low May 26 prices may reflect non-synchronous prices or bid-ask effects of net sales of calls as owners try to avoid having to exercise and acquire the stock just before the dividend date.
The 250-strike put on May 27 had a price of 81.50 when the stock price was 197.7. The 250-strike call had a price of DM .11. The theoretical upper bound for the put price was
.11 − 197.7 + 250 + 21.60 = 74.01.
The actual option price of 81.50 is 7.49 greater than the upper bound. In order to make sense of the price, the tax credit embedded in the option price must be at least 7.49, which implies a credit of 7.49
21.60 = 34.7%.
The day following the dividend, the stock price was 180 with a put price of 72.60, 2.60 above the intrinsic value of 70.45 On June 2 the stock price
was 177.90 and the option price was 73.50, 1.40 above intrinsic value. Even if we assume conservatively that on May 27 the option was mispriced by DM 3, the implied credit is still 4.49, or 21%. Similar patterns are followed by the other put prices in Table 7.46Even with a sizable bid-ask spread, it seems
impossible to explain the prices of Daimler-Benz puts on May 27 without recourse to the tax credit.
Option volume is also consistent with tax credit trading. Daily volume for the 250-strike put, for example, averaged 2762 contracts between May 11 and June 10, and 0 from then until expiration. It was 8.5 times this amount on May 27. Daily volume for the five other puts in Table 7 averaged only 227 contracts over this period.